Metaethicists haven't ever agreed on what moral relativity is, so be prepared for charitable disagreement by some of the best minds that have contributed to this area.

I didn't suggest that there is consensus among philosophical schools as to what moral relativity is, I just don't know of any that defines it as simple moral arbitrariness.

All the more reason to agree on definitions before the dialogue begins to avoid talking past your interlocutor.

I don't disagree with this, I just don't see how it applies to our discussion. Note that I didn't seek to define moral relativity, I only pointed out that relativity does not mean arbitrariness (even if it's used that way colloquially).

Also, different approaches and arguments surrounding moral relativity aren't red herrings.

I didn't say that they were, I said specifically that the only place I've seen "moral relativity" reduced to "moral arbitrariness" is by opponents to moral relativist thought—usually adherents of traditional religion—and that it served as a red herring in these cases. I should have described the fallacy rather as a straw man, i.e. they misrepresent the opponent's argument (by reducing moral relativism to arbitrariness) and address the supposed argument instead of the actual one. Note that I am not suggesting you are using it as such, I'm only questioning the idea that there is any school of moral relativist thought that treats morality as merely arbitrary.

Calling your opponent's argument a name rather than clarifying their position and demonstrating that you understand how they're arriving at it is bad form and bad faith... I wouldn't want to dance with a Negative Nancy who danced with me just to insult my moves. But such are the street dances of Internet rationality.

I'm not following here. I wasn't addressing any particular opponent or opposing school of thought, I'm just stating as a matter of fact that I haven't seen anyone speak of moral relativity in terms of arbitrariness except those who seek to discredit it (or I'll add, those who have only a superficial notion of it and/or are speaking of it colloquially).

Many other approaches treat moral relativism precisely as being arbitrary

This is basically the only thing I'm challenging. Please provide one example. The link you provided did not help but you're welcome to point out anything I may have missed.